It depends on what your new anti-virus found. Was it labeled as "Adware" or something more serious like "Trojan, Malware, etc."
If the detection your new anti-virus was labeled as something like "spyware, trojan or malware" it is something to be more concerned of. If your new anti-virus was able to detect it, then you need to uninstall your old anti-virus and stay away from it. In continuation to what Jim said, each anti-virus has its own definitions list and what it particularly looks for and recognizes as a threat.
There are security softwares that are specialized in certain types of threats. Malwarebytes focuses on malware, and may not detect other forms of viruses. Super Anti-Spyware is used to look for spyware, but is also pretty good at detecting other types of viruses. Other all-around anti-virus softwares like BitDefender and Webroot are on the top list of best anti-viruses. Webroot stands out due to their method of keeping up to date, which is all done through the cloud, which means you never need to manually update or scan, everything is done silently in the background, and is very accurate in what it detects.
When it comes down to it, no anti-virus will find "everything". There are some that will find things quicker, and handle them better than others, but there is no perfect anti-virus. You'll need to do your part and just pay attention to shady looking websites, or suspicious activity on your computer. Don't open attachments in your email from anyone that you do not know, or didn't expect, things like that.
I would never suggest having two all-around anti-viruses, since they contain firewalls, some have a tendency to see each other as a threat. I will however, always suggest having two forms of anti-virus. A premium anti-virus like BitDefender, Webroot, Norton, etc. in mixture with a specialty protection like Malwarebytes or Super Anti-Spyware. They will not interfere with each other, and will provide you with overall better protection.
I hope this helps!